Hypothesis Testing Calculator

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Example

Created on 2024-06-20Asked by Michael Thompson (Solvelet student)

Perform a hypothesis test at the 5% significance level to determine if the mean of a population is different from 50, given a sample mean of 52, a sample standard deviation of 4, and a sample size of 30.

Solution

To perform a hypothesis test at the 5% significance level to determine if the mean of a population is different from 50: First, state the null and alternative hypotheses:

H_0: \mu = 50

H_1: \mu \neq 50

Given: - Sample mean (

\bar{x}

) = 52 - Sample standard deviation (

s

) = 4 - Sample size (

n

) = 30 Calculate the test statistic using the formula for the t-test:

t = \frac{\bar{x} - \mu}{s / \sqrt{n}}

t = \frac{52 - 50}{4 / \sqrt{30}}

t = \frac{2}{4 / \sqrt{30}}

t = \frac{2 \sqrt{30}}{4}

t = \frac{\sqrt{30}}{2}

t \approx \frac{5.477}{2}

t \approx 2.7385

Determine the critical t-value for a two-tailed test at the 5% significance level with

n - 1

degrees of freedom:

\alpha = 0.05

\text{Degrees of freedom} = 30 - 1 = 29

Using a t-table or calculator, the critical t-value is approximately:

t_{\alpha/2, 29} \approx 2.045

Compare the test statistic to the critical t-value:

|t| = 2.7385

t_{\text{critical}} = 2.045

Since

|t| > t_{\text{critical}}

, we reject the null hypothesis. Therefore, there is sufficient evidence at the 5% significance level to conclude that the mean of the population is different from 50. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Logan King on Solvelet

1. Perform a hypothesis test to determine whether the mean weight of a sample of

100

apples is significantly different from

150

grams with a significance level of

\alpha = 0.05

,2. Determine the slope of the tangent line to the curve defined implicitly by the equation

x^3 + y^3 = 9

at the point

(2, 1)

DefinitionHypothesis Testing is a statistical method to make an inference or conclusion about the population on the basis of the sample data. This entails creating a null (H0) and an alternative (H1) hypothesis and using sample data to see if we have enough evidence to reject the null hypothesis. Hypothesis testing is one of the most important tools in scientific research (corresponding to the behavioural attainment of intelligent beings which are learned from observations as well as that of the practice of a design procedure), business decision making, and control the quality of products. Example: if to check the effectiveness of a new drug researchers state H0: Drug has no effect, and H1: Drug has an effect After that, the data is analyzed by means of a statistical test, t-test or chi-square test, and a decision is made.

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